Teaching through Problem Solving

Teaching through Problem SolvingAuthor(s): Cos D. Fi and Katherine M. DegnerSource: The Mathematics Teacher, Vol. 105, No. 6 (February 2012), pp. 455-459Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/10.5951/mathteacher.105.6.0455 .

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Vol. 105, No. 6 • February 2012 | MatheMatics teacher 455

The patterns studied by mathematicians are, for all practical purposes, as real as the atomic particles studied by physicists.

—Mathematical Sciences Education Board

eaching through Problem Solving (TtPS) is an effective way to teach mathemat-ics for understanding. It also provides students with a way to learn mathemat-ics with understanding. In this article,

we present a definition of what it means to teach through problem solving. We also describe a profes-sional development vignette that exemplifies the teaching moves discussed here. Finally, we reflect about TtPS as a way to achieve equity in the math-ematics classroom.

WHAT IS TEACHING THROUGH PROBLEM SOLVING (TTPS)?For the purposes of this article, we define TtPS as pedagogy that engages students in problem solving as a tool to facilitate students’ learning of important mathematics subject matter and mathematical prac-

tices. To illustrate, we provide a sequence of teach-ing moves that encapsulates the definition and the use of TtPS. However, we first want to share with readers our ultimate agenda: We argue that equita-ble pedagogy (NCTM 2000) empowers students to be able to make decisions about their life trajectory.

We argue that mathematical empowerment (a component of educational empowerment) provides opportunities, space, and support to students to develop competencies and dispositions so that they might capitalize on and maximize their learning to make life choices. One such empowering tool is fluency in mathematics and mathematical think-ing (Mason, Burton, and Stacey 1985; Pólya 1945; Schoenfeld 1985). TtPS provides the space for stu-dents to develop such mathematical flexibility. At a local level, TtPS allows all students to enter into the mathematics at hand and enables them to travel through a progressively formalized trajectory that lays out and makes public mathematical practices (solving problems, abstracting, inventing, proving) (Mathematical Sciences Education Board 1990; Romberg 1983).

TtPS is an approach to teaching mathematics that provides students with a way to learn mathematics with understanding.

cos D. Fi and Katherine M. Degner

ProblemTeaching

Solving

Copyright © 2012 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.



456 MatheMatics teacher | Vol. 105, No. 6 • February 2012

This article revisits the idea of equity through TtPS in mathematics education and translates it into immediately usable practice. The fundamental notion is that students come to learn mathemat-ics with understanding through problem solv-ing (Hiebert, Carpenter, et al. 1997; Hiebert and Wearne 2003; Mason, Burton, and Stacey 1985; NCTM 2000; Pólya 1945; Schoen and Charles 2003; Schoenfeld 1985; Hiebert, Gallimore, et al. 2003).

TEACHING MOVESThe use of tasks within pedagogies of sense making and problem solving can adhere to the following general pedagogical moves. The roles of teacher and student are embedded in the teaching moves. What follows is a brief account of these teaching moves; a fuller account is presented in the appendix at www.nctm/mt012.

Move 1Pose the whole task; facilitate students’ under-standing of the problem. A caveat: Do not remove the mathematical complexities of the problem. Do not break down the problem so that students will not have to struggle productively in generating understanding.

Move 2Allow time for students to explore the messiness of the problem, generate conjectures, and build under-standing. When the students share their work, the teacher should consider sequencing students’ work, starting from the intuitive-concrete approaches and strategies and progressing to the more abstract approaches. The starting point matters, in the lens of equity, because a starting point that allows all students to enter into the discussion of the math-ematics affords students better opportunities to engage with the mathematics and its progression. This notion of progressive formalization (Brans-ford, Brown, and Cocking 2000) is a most impor-tant component of TtPS.

Move 3Focus on the big idea of the mathematics. Students and teacher should make the big ideas explicit and public and reason about them and from them.

Move 4Make ideas visible by capturing and recording indi-vidual and group contributions for public examina-tion. Focus on using mistakes to reveal difficulties, misconceptions, and reasoning. Support interactiv-ity that encourages students to step out of their comfort zones. Students should realize that making mistakes and learning from mistakes are part of the process (Shulman 2005).

Move 5Closure: Provide time for students to reflect indi-vidually and privately on what they have learned. Part of this move can be assignments to help stu-dents reinforce, elaborate, and extend understand-ing. Mathematical understanding develops over time through articulated and coherent mathemati-cal struggles negotiated in small steps.

These five teaching moves are exemplified using the chessboard reward task (Fey and Phillips 2005). See figure 1.

TACKLING THE TASKHere we recount an enactment of the task in an in-service mathematics teachers’ professional develop-ment setting. The following vignette from a profes-sional development instructor for secondary school mathematics teachers shows what it means to teach through problem solving.

Move 1Pose the Chessboard Reward task. Allow teachers to make sense of the task (see fig. 1).

Focus on the problem and not the noise (such as the physical chessboard; peasantry, king, queen, servitude, or rubas versus dollars and cents).

Fig. 1 this problem, used in a professional development

setting, serves as the vehicle for the discussion of teaching

moves.

Fig. 1 this problem, used in a professional development

setting, serves as the vehicle for the discussion of teaching

moves.

Chessboard Reward TaskA king and queen want to reward a faithful peasant. The king suggests placing 1 ruba on the first square of a chessboard, 2 rubas on the second square, 4 rubas on the third, 8 on the fourth, and so on until all 64 squares are cov-ered The queen suggests placing 10 rubas on the first square, 25 on the second square, 40 on the third square, 55 on the fourth square, and so on. The servant receives the amount of money on the last square. How many rubas will be placed on the last square of the chessboard under each

plan? Which plan is the better plan and why?

Adapted from Fey and Phillips

(2005)




Move 2Allow time for individual and interpersonal explo-rations. Teachers spent up to thirty minutes gener-ating meaningful connected representations such as a table, or a grid to indicate payments.

Teachers generated and recorded the data into tables and computed differences, as shown in tables 1 and 2. The differences represent yn – yn–1, n > 1. The professional development instructor emphasized the added need to consider xn – xn–1, n > 1, because the average rate of change is

∆∆yx

y y

x xn n

y yn n

y y

n n

=y y−y y

x x−x x−1

−−−1

,

and not just ∆y. As an example, some high school students run the risk of concluding, by focusing only on ∆y, that the relation given by the ordered pairs {(1, 2), (4, 4), (5, 6), (20, 8), (22, 10)} is linear.

Move 3The teachers discussed the differences between the king’s and queen’s payment data. In the queen’s data, they observed that the first rate of change (difference) is a nonzero constant. Hence, they conjectured that the queen’s data are linear. The conjecture was tested by considering the constancy (predicated on similar right triangles) of slopes of lines. We also used the formula for computing the slope of lines, using an arbitrary ordered pair (x, y) and (1, 10) to generate the equation of the line:

−−

= → −( ) → =101

15= →15= → 10 15 1 1)1 1) → =1 1→ = 5 5−5 5−yx

y x− =y x− = (y x(10y x10− =10− =y x− =10− = 15y x15 y x→ =y x→ =1 1y x1 1→ =1 1→ =y x→ =1 1→ = 5 5y x5 5

The professional development instructor and the teachers explored the context significance of the

y-intercept with the following question: What form of the equation of a line is the rule y = 15x – 5? The discussion brought to the surface a potential mis-conception of the y-intercept as the starting point. After considering the practical domain of the task, the teachers concluded that graphically the y-inter-cept at (0, –5) did not make sense in the problem context. However, as a formula, the rule tells us that the queen’s payments are 5 fewer than integer multiples of 15. The authors encourage the reader to consider the question: What other connections do you see arising from such an exploration?

Move 4The constancy of the first rate of change of the queen’s payments contrasts with the rates of change of the king’s payments. The teachers observed that at each stage the rates of change of the king’s data are self-repeating and that the rates of change are exponential. They also argued that there would not be a last column in the finite difference table as they consider subsequent differences in the king’s data. Hence, they concluded that the notion of finite differ-ences does not apply to exponential relations because it appears that there is not a finite number of steps that can yield a nonzero constant rate of change.

The teachers came up with a general rule for the nth-day payment by the king as 2(n–1), where nrepresents a day number between 1 and 64. The rule emerged after serious discussions and debates, questioning peers, mathematical justifications, and mathematical representations. Some teachers used a scatter plot of the data to reconcile visually the growth pattern of the king’s payments. They used a graphing calculator (TI 84: ExpReg) to compute an exponential repression equation. The regression equation was f(x) = 0.5(2x).

Table 1 Queen’s Payment Plan

Day No.

Queen’s Payments

1st Difference

1 10

2 25 15

3 40 15

4 55 15

5 70 15

6 85 15

7 100 15

8 115 15

9 130 15

10 145 15

Table 2 King’s Payment Plan

Day No.King’s

Payments1st

Difference2nd

Difference

1 1

2 2 1

3 4 2 1

4 8 4 2

5 16 8 4

6 32 16 8

7 64 32 16

8 128 64 32

9 256 128 64

10 512 256 128



458 MatheMatics teacher | Vol. 105, No. 6 • February 2012

At this juncture, the professional development instructor asked: “Explain why—or why not—the two expressions 2(n–1) and 0.5(2x) are equivalent.” This question led to a consideration of properties of powers. The teachers shaped and molded the discourse, with minor direction from the profes-sional development instructor. The instructor recorded the teachers’ work on the chalkboard and chart paper as an explicit and public record of the mathematics (Hiebert and Grouws 2007) intended to promote visibility and accountability (Shulman 2005).

A possible extension could be an examination of rates of change of the dependent quantity over smaller equal intervals of the independent quantity. One can begin the discussion via slope:

+ −0 1+ −0 1+ −( .+ −( .+ −0 1( .0 1+ −0 1+ −( .+ −0 1+ −) (+ −) (+ −f x( .f x( . f x) (f x) ( −−−+ − −

0 10 1+ −0 1+ − 0 1

. )0 1. )0 1( .+ −( .+ −0 1( .0 1+ −0 1+ −( .+ −0 1+ −) (+ −) (+ −) ( . )0 1. )0 1x x+ −x x+ −0 1x x0 1+ −0 1+ −x x+ −0 1+ −( .x x( .+ −( .+ −x x+ −( .+ −0 1( .0 1x x0 1( .0 1+ −0 1+ −( .+ −0 1+ −x x+ −0 1+ −( .+ −0 1+ −) (x x) (+ −) (+ −x x+ −) (+ −

(see Coxford et al. 2001). The intent of such discus-sion is to forestall potential misconceptions about

rates of change of exponential functions. More-over, the extension can proceed into a discussion of limits and the derivative of the general family of functions y = bx—here, x ∈ .

Discussions about the better payment plan between the queen and the king also brought out additional mathematical reasoning.

Move 5Teachers chose the king’s model if they put themselves in the peasant’s shoes and consid-ered an employment period longer than seven

days. However, when the employment period is within a one-week period, the queen’s

reward system became the plan of choice for the receiver of the remuneration.

Other intended mathematical residues emerged from the dis-cussion of the prompt, “What is the mathematics embedded in the task?” These new insights

related to the following:

• Rates of change of linear versus exponential relations

• Multiple connected and meaningful representa-tions (verbal, equations, tables, including those for finite differences, and graphs)

• Mathematical practices of problem solving, reasoning, and communicating mathematical reasoning and mathematical thought

A caveat is necessary here: The rate of change was computed on the sequences as shown in

tables 1 and 2. In table 2, the rates of change seem to imply that y′ = y; this would have been the case if the base were e. Another potential incorrect overgeneralization could be a claim that the rates of change of families of functions reside in that family of func-tions. Such was the case for the func-tions used to model the chessboard reward task. However, the deriv-atives of functions do not have to be of the same family as the function. That is, f ′does not have to be in the same family of functions as f. A case in point is the logarithmic functions log x, whose derivative is an algebraic function.

EXAMINING THE PEDAGOGY After the conclusion of the discussion of the mathematics and after having answered the ques-tions of the task, the teachers and the professional development instructor turned their attention to understanding the pedagogy that was used. The instructor used this moment of reflecting on the pedagogy to make public the pedagogy of TtPS. The instructor also wanted to make the argument that TtPS is a robust approach to achieving equity in the classroom and that it can achieve depth of under-standing without sacrificing breadth (that is, the intended curriculum can still be covered).

Basically, explanation of the pedagogy of TtPS involved stepping through the teaching moves as delineated in the previous section. Each move was made public by capturing its name and purpose on chart paper, which was displayed in a sequence that exhibited a pedagogical storyline of TtPS.

The aforementioned vignette from a profes-sional development instructor for secondary school mathematics teachers shows what it means to teach through problem solving. The instructor used TtPS as a way to exemplify to teachers that TtPS should not be an activity “added on” to the beginning or the end of the unit. TtPS is a means of teaching important content.

Teaching important content in this way gives every student in every teacher’s classroom a chance to work with the mathematics instead of watching “experts” do all the work. The idea of letting each student struggle—productively— through math-ematics and engage in mathematical practices is the meaning of the Equity Principle as outlined by NCTM. It is also a responsible way to ensure that students experience the joy, complexity, and beauty of mathematics. Let’s inspire students with the love of mathematics and invite them to partake in math-ematical practices.




BIBLIOGRAPHYBransford, John D., Ann L. Brown, and Rodney R.

Cocking, eds. 2000. How People Learn: Brain, Mind, Experience, and School. Washington, DC: National Academy Press.

Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Harold L. Schoen, Eric W. Hart, Brian A. Keller, Ann Watkins, Beth E. Ritsema, and Rebecca K. Walker. 2001. Contemporary Mathematics in Con-text: A Unified Approach. Course 4, Part A. Chicago: Everyday Learning.

Fey, Jim T., and Elizabeth D. Phillips. 2005. “A Course Called Algebra I.” In Developing Students’ Algebraic Reasoning Abilities, edited by Carole Greens and Carol Findell, pp. 33–45. Lake-wood, CO: National Council of Supervisors of Mathematics.

Garden, Robert A., Svein Lie, David F. Robitaille, Carl Angell, Michael O. Martin, Ina V.S. Mullis, Pierre Foy, and Alka Arora. 2006. TIMSS Advanced Assessment Framework. Boston: TIMSS and PIRLS International Study Center.

Hiebert, James, Thomas P. Carpenter, Elizabeth Fennema, Karen C. Fuson, Diana Wearne, Hanlie Murray, Alwyn Olivier, and Piet Human. 1997. Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann.

Hiebert, James, Ronald Gallimore, Helen Garnier, Karen B. Givvin, Hilary Hollingsworth, Jennifer Jacobs, Angel M. Chui, Diana Wearne, Margaret Smith, Nicole Kersting, Alfred Manaster, Ellen Tseng, Wallace Etterbeek, Carl Manaster, Pat-rick Gonzales, and James Stigler. 2003. Teaching Mathematics in Seven Countries: Results from the TIMSS Video Study. Washington, DC: U.S. Depart-ment of Education, National Center for Education Statistics.

Hiebert, James, and Douglas A. Grouws. 2007. “The Effects of Classroom Mathematics Teaching on Stu-dents’ Learning.” In Second Handbook of Research on Mathematics Teaching and Learning, edited by Frank K. Lester, pp. 371–404. Reston, VA: National Council of Teachers of Mathematics.

Hiebert, James, and Diana Wearne. 2003. “Develop-ing Understanding through Problem Solving.” InTeaching Mathematics through Problem Solving: Grades 6–12, edited by Harold L. Schoen and Ran-dall I. Charles, pp. 3–12. Reston, VA: National Council of Teachers of Mathematics.

Mason, John, Leone Burton, and Kaye Stacey. 1985. Thinking Mathematically. Wokingham, UK: Addi-son Wesley.

Mathematical Sciences Education Board. 1990. “Reshaping School Mathematics: A Philosophy and Framework for Curriculum.” Washington, DC: National Academy Press.

National Council of Teachers of Mathematics

(NCTM). 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

Pólya, George. 1945. How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press.

Romberg, Tom A. 1983. “A Common Curriculum for Mathematics.” In Individual Differences and the Common Curriculum: Eighty-Second Yearbook of the National Society for the Study of Education, Part 1,edited by G. D. Fenstermacher and J. I. Goodlad, pp. 121–59. Chicago: University of Chicago Press.

Schoen, H. L., and Randall I. Charles. 2003. Teach-ing Mathematics through Problem Solving: Grades 6–12. Reston, VA: National Council of Teachers of Mathematics.

Schoenfeld, Alan H. 1985. Mathematical Problem Solv-ing. San Diego, CA: Academic Press.

Shulman, Lee. 2005. The Signature Pedagogies of the Professions of Law, Medicine, Engineering, and the Clergy: Potential Lessons for the Education of Teach-ers. http://hub.mspnet.org/index.cfm/11172.

Stein, Mary K., Margaret S. Smith, Marjorie A. Hen-ningsen, and Edward A. Silver. 2000. Implement-ing Standards-Based Mathematics Instruction: A Casebook for Professional Development. New York: Teachers College Press.

COS D. FI, superfi @me.com, is an independent consultant on teacher professionalization and embodying mathematics teaching at Seven 7 Growing the World. He is interested in effective pedagogies of mathematics and mathematics teachers’ profes-sional learning.

KATHERINE M. DEGNER, [email protected], is a doctoral student at the University of Iowa. She is interested in gifted education in math-ematics and speaks at a variety of conferences.

a fuller account of the fi ve pedagogical moves of teaching through Problem

solving is available by downloading one of the free apps for your smart-phone and then scaning this tag to access www.nctm.org/mt012.



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Teaching through Problem Solving